As StoneyB points out, that is acting as a relative pronoun here, giving a sense like “heath that the wind whirls away” or “heath which the wind whirls away”.
As FumbleFingers points out, the two lines are from a poem: Fallen is Thy Throne by Thomas Moore
(born 1779, 36 years before Charlotte Brontë).
I don't know the original publication date of Fallen is Thy Throne, except that apparently its first wide availability was in Moore's Sacred Songs  in May or June, 1816. Thus the “wild wind whirls away” quotation may be an anachronism: I've supposed the time frame of Jane Eyre to be ca. 1809 or 1810, based on reference in Chapter XXXII to Scott's Marmion (late 1808) as “a new publication”:
“... I have brought you a book for evening solace,” and he laid on the table a new publication—a poem ...
While I was eagerly glancing at the bright pages of “Marmion” (for “Marmion” it was), St. John stooped to examine my drawing. ...
 A Series of Sacred Songs, Duetts and Trios, The Words by Thomas Moore, Esqr. The Music, Composed and Selected by Sir John Stevenson, part 1 (London: J. Power / Dublin: William Power, 1816; Philadelphia: Published by Geo. E. Blake, 1817?); part 2 (London: J. Power, 1824).